Question

A 135 -g sample of a metal requires \(2.50 \mathrm{~kJ}\) to change its temperature from \(19.5^{\circ} \mathrm{C}\) to \(100.0^{\circ} \mathrm{C}\). What is the specific heat of this metal?

Short answer

The specific heat capacity of the metal is approximately \(0.23 \frac{J}{g \cdot ^\circ C}\).

Step-by-step solution

Understand the problem

The exercise provides the mass of the metal, the amount of heat energy absorbed, and the change in temperature. Our goal is to calculate the specific heat capacity of the metal, which is the amount of heat energy required to raise the temperature of 1 gram of the substance by 1 degree Celsius.

Write down the formula for specific heat capacity

The formula to calculate the specific heat capacity (c) is given by: \(c = \frac{q}{m \Delta T}\), where \(q\) is the heat absorbed (in joules or kilojoules), \(m\) is the mass of the substance (in grams), and \(\Delta T\) is the change in temperature (in degrees Celsius or Kelvin).

Convert heat energy into the correct unit

Ensure that the heat energy absorbed is in joules. Since we have been given the heat in kilojoules, we convert it to joules by multiplying by 1000. Consequently, \(q = 2.50 \mathrm{~kJ} = 2500 \mathrm{~J}\).

Calculate the change in temperature

The change in temperature (\(\Delta T\)) is the final temperature minus the initial temperature. Hence, \(\Delta T = 100.0^\circ \mathrm{C} - 19.5^\circ \mathrm{C} = 80.5^\circ \mathrm{C}\).

Calculate the specific heat capacity

Substitute the values for \(q\), \(m\), and \(\Delta T\) into the formula to calculate the specific heat capacity (c). Thus, \(c = \frac{2500 \mathrm{~J}}{135 \mathrm{~g} \times 80.5^\circ \mathrm{C}}\).

Perform the calculation

Solve the equation from the previous step to find the specific heat capacity. \(c = \frac{2500 \mathrm{~J}}{135 \mathrm{~g} \times 80.5}\), which simplifies to \(c = \frac{2500 \mathrm{~J}}{10867.5 \mathrm{~g} \cdot \mathrm{C}}\).

Simplify to find the specific heat capacity

Divide the heat energy by the product of mass and change in temperature to get the specific heat capacity: \(c = \frac{2500}{10867.5} \approx 0.23 \frac{J}{g \cdot ^\circ C}\).

Key concepts

Thermal Energy Transfer

Thermal energy transfer, also known as heat transfer, is the movement of heat from one place to another. Heat can be transferred in three ways: conduction, convection, and radiation. In solids, heat transfer usually occurs through conduction, where heat moves from an area of higher temperature to an area of lower temperature through direct contact between molecules.

In the exercise provided, when a piece of metal absorbs thermal energy, the molecules within the metal start to move faster and the temperature of the metal increases. This transfer of energy is particularly important to note because specific heat capacity is a property that tells us how much energy is needed to raise the temperature of a material.

The transfer of thermal energy is crucial for determining specific heat because it considers the amount of energy needed to raise the temperature of a specific mass of a substance. Thus, understanding how thermal energy transfer works helps us to comprehend why different substances require different amounts of energy to change their temperature by the same amount.

Temperature Change Calculations

Temperature change calculations involve determining the difference in temperature as a substance absorbs or releases heat. The change in temperature, symbolized as \(\Delta T\), is a crucial component in calculating the specific heat capacity of a substance.

In the exercise, to find \(\Delta T\), we subtract the initial temperature of the metal from its final temperature after it has absorbed heat. Here is the mathematical representation of the temperature change:\[\Delta T = T_{final} - T_{initial}\]
For our specific example, the final temperature is \(100.0^\circ \mathrm{C}\) and the initial temperature is \(19.5^\circ \mathrm{C}\), giving us \(\Delta T = 100.0^\circ \mathrm{C} - 19.5^\circ \mathrm{C} = 80.5^\circ \mathrm{C}\).

Why is Temperature Change Important?

• It helps understand how much a substance responds to a given amount of energy.
• It allows us to apply the specific heat capacity formula correctly to find out the energy required or released during temperature changes.

By accurately calculating temperature change, we can directly link the thermal energy transfer to the resultant increase or decrease in temperature.

Heat Capacity Formula

The heat capacity formula is critical for calculating the specific heat capacity of substances. The specific heat capacity (\(c\)) is defined as the amount of heat (\(q\)) required to raise the temperature of a unit mass (\(m\)) of a substance by one-degree Celsius (or one Kelvin).

The formula to find specific heat capacity is:\[c = \frac{q}{m \Delta T}\]
Where:\[q\] = heat absorbed or released in joules (J) or kilojoules (kJ)
\[m\] = mass of the substance in grams (g)
\[\Delta T\] = change in temperature, calculated as final temperature minus initial temperature

Using this formula, we can solve the problem provided in the exercise by substituting the appropriate values. First, we converted the given heat from kilojoules to joules for consistency with the mass in grams and temperature change in Celsius. After determining the values of \(m\) and \(\Delta T\), we applied them to the formula.

Importance of the Specific Heat Capacity Formula

• It allows us to calculate the energy required to heat different materials and compare their thermal properties.
• It is essential in various applications, from designing heating systems to understanding environmental physics.

With the formula, we learn not only how substances interact with heat but also how to engineer solutions that require precise temperature control.